For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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-1
votes
2answers
235 views

Symmetric matrix and inner product

If A is real, symmetric, regular, positive definite matrix in $R^{n.n}$ and $x,h\in R^n$, why is it $\langle Ah,x\rangle = \langle h,A^T x\rangle =\langle Ax,h\rangle$? Is there some rule or theorem ...
1
vote
1answer
1k views

Find formulas for the entries of $M^n$, where $n$ is a positive integer ($2\times 2$ matrix)

I got the following solution by doing: $P^{-1}\times M^n\times P$ but it is not correct. I spent hours trying to solve this and find help online but I am out of options. Please help :( Note: ...
2
votes
1answer
113 views

How prove that $\;(1-\mathrm{Tr}\,A)^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4\;\;?$

Let $A=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{bmatrix}$ be an orthogonal matrix with $a_{i,j}\in \mathbb R$, where $\det(A)=1$ ...
1
vote
0answers
62 views

Singular values of $A$ and eigenvalues of $\left[\begin{smallmatrix} 0 & A \\ A^T & 0 \end{smallmatrix}\right]$.

(Roger p.418) Let $A$ be $m \times n$ matrix, $q=\min(m,n)$, and $B=\begin{bmatrix}0&A\\A^*&0\end{bmatrix}$. Let $\sigma_1 ,\ldots,\sigma_q \ge 0$. The singular values of $A$ are ...
1
vote
1answer
303 views

Is my row calculation of row echelon form correct?

I was directed by a community member to a resource on how to calculate the row echelon form of a matrix here. The resource says: ...
1
vote
1answer
114 views

Minimal spectral radius of a primitive matrix

Given the set of all primitive matrices of dimensions $m$ by $m$ that are non-negative and integer - which one is the matrix with the minimal spectral radius? Edit (according to the first comment): ...
1
vote
1answer
79 views

Dimension of endomorphisms subspaces

Let $V$ an $\mathbb R$-vector space with finite dimension $n$ ($n \neq 0$). We know that all endomorphisms $f$ on $E$ can be written as a linear combination of some projectors of $E$. Otherwise: if ...
1
vote
1answer
820 views

What is matrix reduction to normal form PAQ?

Here is my university syllabus. I started doing math in vacation just to get a head start because I am a dunce in math. So, I began with chapter 2 - matrices - because it looked easier. I went half ...
1
vote
1answer
46 views

Finding SVD from unit eigen values

Suppose $A$ is a 2 by 2 symmetric matrix with unit eigenvectors $u_1$ and $u_2$. If its eigenvalues are $\lambda_1=3$ and $\lambda_2=-2$, what are the matrices $U,\Sigma,V^T $ in its SVD? How to do ...
2
votes
2answers
153 views

Computing positive semidefinite (PSD) rank with mathematical software

I would like find positive semidefinite (PSD) rank or a decomposition for 10~15 size square nonnegative matrices with the aid of some mathematical software. I wonder if it is feasible with standard ...
3
votes
1answer
175 views

For given $n\times n$ matrix $A$ singular matrix, prove that $\operatorname{rank}(\operatorname{adj}A) \leq 1$

For given $n\times n$ matrix $A$ singular matrix, prove that $\operatorname{rank}(\operatorname{adj}A) \leq 1$ So from the properties of the adjugate matrix we know that $$ A \cdot ...
3
votes
3answers
785 views

Calculating the minimal polynomial of this matrix

Given $\begin{pmatrix} 1 & 1 & \cdots & 1 \\ 2 & 2 & \cdots & 2 \\ \cdot & \cdot & \cdots & \cdot \\ n & n & \cdots & n \end{pmatrix}$, calculate ...
1
vote
1answer
341 views

If modulus of each one of eigenvalues of $B$ is less than $1$, then $B^k\rightarrow 0$

Let $B$ be a $n\times n$ matrix and let $X$ be the set of all eigenvalues of $B$. Prove that if $|m|<1$ then $\lim \limits_{k\rightarrow\infty}B^k=0$, where $m=\max X$. Thanks. Actually, there ...
1
vote
2answers
67 views

Am I understanding vectors and matrices properly?

So, here is my understanding of a Vector: A vector is an ordered set of real numbers that lie in the space $R^n$ where $n$ is the size of the vector. So if ...
0
votes
1answer
81 views

Quadratic matrices: When is $A^\top B^\top = AB$?

When is $A^\top B^\top = AB$ true? Context: I saw this as $AB=BA \implies (AB)^\top=(BA)^\top=A^\top B^\top=AB$.
2
votes
2answers
142 views

Matrices with at most one negative eigenvalue

Suppose a vector $y$ and a symmetric matrix $M$ are given. \begin{equation} \forall x; \quad x^Ty=0 \implies x^TMx \ge 0 \end{equation} Prove that $M$ has at most one negative eigenvalue.
0
votes
1answer
285 views

Solve linear congruence system of equations using Gaussian Elimination

I was solving system of linear congruence equations, let me put it this way: there are n variables represented as X, the solution of X must be integers, ...
3
votes
1answer
74 views

Hamiltonian Quaternions: A Call for Counterexample for ($AB=I_m \implies n≥m$)

How can I build a counterexample to $AB=I_m \implies n≥m$ in the ring of Hamiltonian quaternions? Notice that the vectors $(1,i)$ and $(j,k)=(1,i)j$ are linearly dependent as vectors of the right ...
0
votes
1answer
409 views

Matrix norm of a normal matrix

A normal matrix defined over a complex vector space has the property, that $\|A\|_2$ is its largest eigenvalue and now I was wondering whether this is also true for matrices defined over the real ...
3
votes
2answers
528 views

Why does the $n$-th power of a Jordan matrix involve the binomial coefficient?

I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$-th power is: $$ J_k(\lambda)^n = \begin{bmatrix} \lambda^n & \binom{n}{1}\lambda^{n-1} & ...
1
vote
1answer
228 views

Example of Matrix in Reduced Row Echelon Form

I'm struggling with this question and can't seem to come up with an example: Give an example of a linear system (augmented matrix form) that has: reduced row echelon form consistent 3 equations 1 ...
0
votes
1answer
1k views

Finding the probability from a markov chain with transition matrix

Consider the Markov Chain with state space $S=\{v,w,x,y,z\}$, transition matrix below: $$\left[\begin{array}{cccccccccc} 0 & 0.4 & 0.6 & 0 & 0\\ 0 & 0.5 & 0.5 & 0 & ...
0
votes
1answer
137 views

Derivative of Trace of Matrix wrt parameters

I have the following function which I need to find the derivative of $$L=trace(\Sigma K^{-1})$$ where $K$ is a function of $\theta$ and $\Sigma$ is constant. If I'm correct what I need to do to find ...
0
votes
0answers
173 views

solving laplace in 3d using finite differences

I have created a function laplace3d() which accepts a 3D array describing the boundary conditions and -Inf at the places of unknown. It then calculates these ...
4
votes
1answer
132 views

Understanding an algorithm for computing a matrix polynomial

I'm trying to understand this algorithm by Charles Van Loan for evaluating a matrix polynomial $p(\mathbf A)=\sum\limits_{k=0}^q b_k \mathbf A^k=b_0\mathbf I+b_1\mathbf A+\cdots$ (where $\mathbf A$ is ...
5
votes
3answers
190 views

How to construct a $2\times 2$ real matrix $A$ not equal to Identity such that $A^3=I$?

How to construct a $2\times 2$ real matrix A not equal to Identity such that $A^3$=I? There is a correspondence between the ring of complex numbers and the ring of $2\times2$ matrices (0 matrix is ...
4
votes
1answer
56 views

Given $T(A) = A^t$ in $M_{n\times n}(\mathbb R)$. Find the polynomials and find if it's diagonalizable

Given the vector space $M_{n\times n} (\mathbb R)$ and a transformation $T(A) = A^t$ (transpose): Find $m_T$, $P_T$ (the minimum polynomial and the characteristic polynomial respectively.) ...
12
votes
4answers
4k views

How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
2
votes
1answer
2k views

Armijo's rule line search

I have read a paper (http://www.seas.upenn.edu/~taskar/pubs/aistats09.pdf) which describes a way to solve an optimization problem involving Armijo's rule, cf. p363 eq 13. The variable is $\beta$ ...
1
vote
1answer
70 views

Whether solutions of a particular matrix equation are only the permutation matrices

Let $V \in R^{d \times d} $ be a real orthogonal matrix. Denote by $V \circ V$ the Hadamard product (elementwise product). I wish to show that if $V \left(V \circ V \right)^T V$ is "close" to $V ...
5
votes
2answers
84 views

When does $A^p$ is a diagonal $2\times 2$ matrix imply that $A$ is a diagonal matrix

Let $A \in\mathrm{SL}(2,\mathbb{C})$ and $p > 1$ be a natural number. Under which conditions the following statement is true? $A^p$ is a diagonal matrix implies $A$ is a diagonal matrix For ...
2
votes
2answers
80 views

If $X=[x_{ij}]_{n \times n}$ then how prove $X^n=0$

Let $n\in \mathbb N$ and $A_1,A_2,..,A_n$ be arbitrary sets. Now define $X=[x_{ij}]_{n \times n}$ where $$x_{ij}= \begin{cases} 1 , & \text{$A_i$$\subsetneq$}A_j \\ 0 , & \text{otherwise} \\ ...
12
votes
2answers
379 views

Product of nilpotent matrices.

Let $A$ and $B$ be $n \times n$ complex matrices and let $[A,B] = AB - BA$. Question: If $A , B$ and $[A,B]$ are all nilpotent matrices, is it necessarily true that $\operatorname{trace}(AB) ...
3
votes
1answer
53 views

Why is $\operatorname{rank}(A^* A)=\operatorname{rank}(A) \Leftrightarrow$ $(A^* Ax=0 \Leftrightarrow Ax=0)$?

Let $A \in M_{m\times n}(F)$ and $x \in F^n$. $A^*$ is the adjoint of $A$. Why is $\operatorname{rank}(A^* A)=\operatorname{rank}(A)$ equivalent to $A^* Ax=0$ if and only if $Ax=0$?
4
votes
5answers
140 views

For which $t \in \mathbb{R}$ is matrix diagonalizable

$\begin{bmatrix} 1 & t & 25 \\ 0 & t & t+1 \\ 0 & 0 & -1 \end{bmatrix}$ I calculated the characteristic polynomial which is $p(\lambda) = (\lambda^2 - 1)(t-\lambda)$. But ...
1
vote
1answer
64 views

Prove that if $\gcd{f, P_A} = 1$ for some matrix $A$ and polynom $f$ then $f(A)$ is invertible

Let $f \in F[x]$ and $A \in M_{n x n} (\mathbb F)$. Prove: If $\gcd\{f,P_A\} = 1 \rightarrow f(A)$ is an invertible matrix. This is what I did so far: If the $\gcd\{f,P_A\} = 1$ then $f$ and ...
3
votes
2answers
145 views

Calculating determinant of matrix

I have to calculate the determinant of the following matrix: \begin{pmatrix} a&b&c&d\\b&-a&d&-c\\c&-d&-a&b\\d&c&-b&-a \end{pmatrix} Using following ...
7
votes
1answer
319 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $A$ be the diagonal matrix w/alternating in sign diagonal entries: $$ A = \begin{pmatrix} (-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\ 0 & ...
1
vote
1answer
48 views

Can't figure out this transformation matrix

So basically I want to write a transformation matrix to take me out of one coordinate system and into another. The transformation has to be as follows: 1) The positive z axis normalized as ...
6
votes
2answers
63 views

Finding the number of symmetric,positive definite $10 \times 10$ matrices having…

I was looking at old exam papers and I was stuck with the following problem: What is the number of symmetric,positive definite $10 \times 10$ matrices having trace equal to $10$ and determinant ...
1
vote
1answer
96 views

Can the diagonal elements of a precision matrix be 0

I have this confusion that why the diagonal elements of the precision matrix cannot be 0? Any suggestions will be much appreciated
1
vote
0answers
57 views

Derivative of a vector with scalar product in denominator

I'm struggling with a partial derivative of the following form: \begin{equation} \frac{\partial}{\partial \vec{x}} \frac{\vec{x}}{\vec{e}_3^T\,\vec{x}}, \end{equation} where $\vec{x} \in ...
1
vote
0answers
394 views

Eigenvalues of Block Anti-Diagonal Matrix

In line with this answer, I am trying to find the eigenvalues of: $\mathbf P\mathbf K\mathbf P^\top=\begin{pmatrix}& d_1 & & & & & & \\d_1 & & e_1 & ...
2
votes
1answer
106 views

Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
4
votes
3answers
283 views

Matrix Inverses

So in class we have been discussing matrix inverses and the quickest way that I know of is to get a matrix A, and put it side by side with the identity matrix, like $[A|I_{n}]$ and apply the ...
0
votes
1answer
211 views

What is the error in Newton's Method for Matrix Inversion?

I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds. Suppose we have, as ...
0
votes
0answers
35 views

Given $P=X^* BX$, how can I solve for $X$? The inertia of $P =$ the inertia of $B$, and $P$ is a diagonal matrix.

I am doing undergraduate research this summer, and I have taken up to advanced linear algebra, but I haven't taken matrix analysis yet. Everything I've learned about it so far I've taught myself in ...
1
vote
1answer
667 views

Find non-singular matrices P and Q such that PAQ is in the normal form for the matrix A.

$A= \left[ \begin{array}{ccc} 1 & 2 & 3 & -2 \\ 2 & -2 & 1 & 3 \\ 3 & 0 & 4 & 1 \end{array} \right]$ $A=IAI$ $\left[ \begin{array}{ccc} 1 & 2 & 3 & -2 ...
2
votes
1answer
135 views

Eigenvectors of matrices which commute with a projection

Just a quick question. Cant seem to prove it or find any relevant references! Maybe it's really simple :\ Is the following statement true (for square matrices of the same finite dimension)? If there ...
0
votes
1answer
380 views

Computing the Frobenius normal form

I was wondering whether someone could give me an example how one actually determines the Frobenius normal form of a given matrix. Further, it seems hard to find an example where the new basis is ...